# Properties

 Label 259182.fh Number of curves $6$ Conductor $259182$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("fh1")

sage: E.isogeny_class()

## Elliptic curves in class 259182.fh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.fh1 259182fh6 $$[1, -1, 1, -14939559779, -702833547127269]$$ $$285531136548675601769470657/17941034271597192$$ $$23170271092499019938343048$$ $$[2]$$ $$314572800$$ $$4.3274$$
259182.fh2 259182fh4 $$[1, -1, 1, -935498939, -10937714017557]$$ $$70108386184777836280897/552468975892674624$$ $$713495986231622458635118656$$ $$[2, 2]$$ $$157286400$$ $$3.9808$$
259182.fh3 259182fh5 $$[1, -1, 1, -318645779, -25146803187525]$$ $$-2770540998624539614657/209924951154647363208$$ $$-271111350310116635073441084552$$ $$[2]$$ $$314572800$$ $$4.3274$$
259182.fh4 259182fh2 $$[1, -1, 1, -98798459, 95018511723]$$ $$82582985847542515777/44772582831427584$$ $$57822356616188051279056896$$ $$[2, 2]$$ $$78643200$$ $$3.6342$$
259182.fh5 259182fh1 $$[1, -1, 1, -76495739, 257212812651]$$ $$38331145780597164097/55468445663232$$ $$71635720864281089015808$$ $$[2]$$ $$39321600$$ $$3.2877$$ $$\Gamma_0(N)$$-optimal
259182.fh6 259182fh3 $$[1, -1, 1, 381058501, 747048148971]$$ $$4738217997934888496063/2928751705237796928$$ $$-3782389016468744260638599232$$ $$[2]$$ $$157286400$$ $$3.9808$$

## Rank

sage: E.rank()

The elliptic curves in class 259182.fh have rank $$1$$.

## Complex multiplication

The elliptic curves in class 259182.fh do not have complex multiplication.

## Modular form 259182.2.a.fh

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2q^{5} - q^{7} + q^{8} + 2q^{10} + 2q^{13} - q^{14} + q^{16} + q^{17} - 4q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.